79
Acta Math. Univ. Comenianae
Vol. LXXX, 1 (2011), pp. 79–102
LAPLACE TRANSFORMS AND SHOUT OPTIONS
G. ALOBAIDI, R. MALLIER and S. MANSI
Abstract. We use Laplace transform methods to examine the optimal exercise
boundary for shout options, which give the holder the right to lock in the profit to
date while retaining the right to benefit from any further upside. The result of our
analysis is an integro-differential equation for the location of this optimal exercise
boundary. This equation is a nonlinear Fredholm equation, or more specifically, an
Urysohn equation of the first kind. Applying an inverse Laplace transform to this
equation allows us to find the behavior of the free boundary close to expiry. The
results are given for both call and put shout options.
1. Introduction
In the past twenty years, the role and the complexity of financial contracts have
grown tremendously, causing a dramatic change in the financial industry. Issuers, investors, and government regulators have increased their reliance on derivative instruments to augment the liquidity of markets, to reallocate financial
risks among market participants, and to take advantage of differences in costs
and returns between these markets. A variety of financial contracts, ranging from
basic American-style vanilla options to more exotic complex contracts, have been
introduced to cater to the needs of a variety of investor profiles. One such financial
contract is a shout option. This is an American-style exotic option which contains
an early exercise feature that enables the holder to lock in the profit to date while
still retaining the right to benefit from any additional upside, and as with any
option carrying early exercise rights, there is an element of uncertainty as to the
actions that the holder will undertake. Shout options are frequently embedded
in other contracts such as the segregated funds sold by Canadian life insurance
companies[8] and protective floor indexes. A brief introduction to shout options
can be found in [41].
When a vanilla American option is exercised, the holder has the right to buy (a
call) or sell (a put) the underlying security at a pre-determined exercise price E.
Like vanilla Americans, shouts can only be exercised when they are in-the-money,
meaning that the stock price is greater than the exercise price for a call or less than
the exercise price for a put. When a shout option is exercised, the holder not only
Received April 18, 2010; revised September 27, 2010.
2001 Mathematics Subject Classification. Primary 44A10, 45G10.
Key words and phrases. American options; shout options; integro-differential equations;
Laplace transforms.
80
G. ALOBAIDI, R. MALLIER and S. MANSI
has the right to buy or sell the underlying security at the exercise price, but also
receives an at-the-money vanilla European option, an at-the-money option being
one whose exercise price is equal to the price of the underlying at that particular
time, with this new option having the same expiry as the original option. Thus
a shout option essentially consists of a vanilla American option together with a
forward start option, the strike price of which is set when the American option
is exercised. In effect, the holder of a shout option has the right to reset the
exercise price of the option, provided that the option is in-the-money, and receive
the difference between the old and the new exercise prices in cash. This enables
the buyer to lock in the profit to date while still retaining the right to benefit from
any further upside.
Since shout options are American-style, their valuation involves determining
whether the option should be exercised prior to expiry, which leads to a free
boundary problem, with the boundary separating the region where early exercise
is beneficial from that where it is not. Early exercise, or shouting, can only occur
when the option is in-the-money. Once the free boundary is hit, shouting occurs
and the option is exchanged for a new (plain vanilla European) option whose
exercise price is the stock price at that particular time, together with a payment
of the difference between the original and new exercise prices. Since this new
option is European, it can be priced using the Black-Scholes formula [7], and it
follows that on the free boundary, the value of a shout option is the value of this
European option together with the value of the payment. It is worth noting that
when a vanilla American option is exercised, the holder receives a payment but no
new option, so the pay-off from early exercise is sweeter for a shout option than for
a vanilla American, and because of this a shout is more likely to be exercised early
than a vanilla American. Although some exotic contracts do exist with multiple
shouting opportunities, in this analysis, we assume that the holder can shout only
once so that there is only one free boundary whose position must be optimized;
with multiple shouting opportunities, there would be multiple free boundaries.
Much of the work done to date on shout options is numerical, although as with
other options involving a free boundary and choice on the part of an investor, some
standard numerical techniques such as the forward-looking Monte Carlo method
are difficult to use because they cannot effectively handle the optimization component of shout options. In [8], a Green’s function approach was used. With this
approach, it is assumed that early exercise could only occur at a limited number
of fixed times t1 < t2 < · · · < tn−1 < tn between the current time t and the expiry
T > t, so that the option is treated as Bermudan-style or semi-American rather
than American-style, and then the value of the option at the time tm is used to
compute the value at time tm−1 , which in turn is used to compute the value at
time tm−2 and so on. The value at time tm−1 is computed using an integral involving the product of the Green’s function with the value at time tm , with this
integral being evaluated numerically. More standard numerical methods, such as
finite differences, have also been applied to shout options [12, 13, 43]. In this
paper, we take an analytical rather than a numerical approach, follow [24, 4], and
use partial Laplace transform methods [16, 11] to derive the integro-differential
LAPLACE TRANSFORMS AND SHOUT OPTIONS
81
equations necessary to locate the optimal exercise boundary. The result for both
call and put shout options is a nonlinear Fredholm integro-differential equation for
the location of the free boundary. An inverse Laplace transform is then applied
to this integro-differential equation, resulting in a second integro-differential equation which allows us to determine the asymptotic behavior of the free boundary
close to expiry. The idea of using integral and integro-differential equations to
tackle American-style option is of course not new, and a number of authors have
used integral equations to study vanilla Americans in the past, including McKean
[26] and Van Moerbeke [42] who considered American calls, with later work including the studies by Kim [21], Jacka [19] and Carr [9], all of whom looked at
the difference between European and American prices, and several recent papers
[39, 23, 17] on the American put. A number of these integral equation approaches
have used integral transforms, notably Fourier transforms [39] and Laplace transforms [24, 22], and the use of these approaches can be traced back to much earlier
work on diffusion problems in physical problems [16, 32], and an overview of this
earlier work and other integral equation formulations of these types of problems
is given in §3.5 of [11].
The remainder of the paper is organized as follows. Section 2 contains our
analysis using Laplace transform methods to locate the free boundary for both
call and put shout options on a dividend paying asset under the Black-Scholes
model, and explains how the analysis for the calls differ from that for the put.
Section 3 contains a discussion of our results.
2. Analysis
2.1. Formulation of the problem
It is well known that the value V (S, t) of a vanilla European option with constant
dividend yield obeys the Black-Scholes-Merton partial differential equation or PDE
[7, 27],
(1)
σ2 S 2 ∂ 2 V
∂V
∂V
+ (r − D) S
+
− rV = 0 ,
∂t
2 ∂S 2
∂S
where S is the price of the underlying and t < T is the time, with T being
the expiry time when the holder will receive the pay-off from the option, which
is max(S − E, 0) for a call with a strike price of E and max(E − S, 0) for a
put. This equation was originally presented by Black & Scholes [7] for options on
stocks without dividends, and later extended by Merton [27] to include a constant
dividend yield. The parameters in the above equation are the risk-free rate, r, the
dividend yield, D, and the volatility, σ, all of which are assumed constant in the
present analysis. To simplify the analysis, we will work in terms of the tenor, or
remaining life of the option, τ = T − t, so that (1) is replaced by
(2)
∂V
σ2 S 2 ∂ 2 V
∂V
=
+ (r − D) S
− rV ,
2
∂τ
2 ∂S
∂S
82
G. ALOBAIDI, R. MALLIER and S. MANSI
While this equation (2) is applicable for τ ≥ 0 for European options, it is also
governs the price of options for which early exercise is permitted, but in those
cases, the equation is only valid where it is optimal to hold the option, and (2)
must be solved together with the appropriate conditions at the optimal exercise
boundary, whose location is unknown and must be solved for. In what follows,
we will label the position of the free boundary as S = Sf (τ ), which we can invert
to give τ = τf (S) as the time at which early exercise should occur. Merton [28]
remarked on the fact that different securities may obey the same equation, and
that it is the boundary and initial conditions which differentiate them, so that
shout, European and American options, along with many others, all obey (2) but
with differing boundary conditions. The pay-off for a shout option held to maturity
is the same as that for an American held to maturity or for a European, namely
max(S − E, 0) for a call and max(E − S, 0) for a put, where E is the original strike
price of the option.
Just as with American options, shout options can be exercised early at any
time provided they are in-the-money, meaning that the stock price is greater than
the exercise price for a call or less than the exercise price for a put. Just as with
American options, this leads to the constraint that the price of the option cannot
fall below the pay-off from immediate exercise, In the case of a shout call, we can
only shout if S > E, and the possibility of shouting leads to the constraint V > Vf
for S > E where Vf is the pay-off from shouting,
 
σ2 √
r
−
D
+
τ
−Dτ
2
Se

√
erfc −
Vf (S, τ ) = S − E +
2
2σ
√ 
(3)
 2
r − D − σ2
τ
S e−rτ
,
√
erfc −
−
2
2σ
this being the difference between the current price of the underlying and the original strike price together with the value of an at-the-money European call, which we
can price using the Black-Scholes formula [7]. In this expression, erfc denotes the
complementary error function. Similarly, for a shout put, we have the constraint
V > Vf for S < E where
√ 

2
r − D + σ2
τ
S e−Dτ

√
Vf (S, τ ) = E − S −
erfc 
2
2σ
(4)


σ2 √
r
−
D
−
τ
−rτ
2
Se
.
√
+
erfc 
2
2σ
As with American options, the possibility of shouting leads to a free boundary
where it is optimal to shout. Several properties of this free boundary are known.
Firstly, we know the value of the option at the free boundary,where V = Vf given
by (3) and (4) above, and also the value of the option’s delta, or derivative of
LAPLACE TRANSFORMS AND SHOUT OPTIONS
83
its value with respect to the stock price, at the free boundary, where (∂V /∂S) =
(∂Vf /∂S), which for a call is
 
 
σ2 √
σ2 √
r
−
D
+
τ
r
−
D
−
τ
−Dτ
−rτ
2
2
e
e
−
,
√
√
(5)
1+
erfc −
erfc −
2
2
2σ
2σ
while for a put it is
(6)
√ 
√ 


2
2
τ
τ
r − D + σ2
r − D − σ2
−rτ
e−Dτ
e
+
.
√
√
−1 −
erfc 
erfc 
2
2
2σ
2σ
The condition on the delta (∂V /∂S) comes from requiring that it be continuous
across the boundary, and is essentially the high contact or smooth-pasting condition, which was first proposed by Samuelson [37] for American options.
Secondly, we know the location of the free boundary at expiry τ = 0 is
Sf (0) = E, which can be deduced intuitively because the pay-off for early exercise is so sweet for shout options. In our terms, τf (E) = 0. We also know that
the optimal exercise boundary moves upwards (or at worst is flat) as we move
away from the expiration date for a call, and downwards (or again at worst is flat)
for a put.
Thirdly, we know the location of the free boundary as τ = T − t → ∞ from
the behavior of the perpetual shout option, for which the pay-off from shouting is
the rebate together with an at-the-money perpetual European option, and since
the value of a European tends to zero as the duration tends to infinity (assuming
the dividend yield is non-zero), it follows that for a non-zero dividend yield, a
perpetual shout behaves like a perpetual American, so that as τ → ∞,
Sf (τ ) → S∗ =
(7)
α=
E
,
1 − 1/α
i
p
1 h 2
σ − 2(r − D) ± 4(r − D)2 + 4(r + D)σ 2 + σ 4 ,
2
2σ
where we take the + sign for a call and the − sign for a put, so that S∗ > E for a
call and S∗ < E for a put. In our terms, τf (S) → ∞ as S → S∗ . A put without
dividends also falls within the above framework, but for a call option without
dividends, as τ → ∞, the value of the option at the boundary tends to 2S − E,
while the value of the option below the boundary is S, and we deduce that in the
absence of dividends, the optimal exercise boundary for a call is flat and located
at Sf (τ ) ≡ E.
For a shout call with a non-zero dividend yield, the optimal exercise boundary
will lie between these two limits, E ≤ Sf (τ ) ≤ S∗ , and of course early exercise is
optimal if S ≥ Sf (τ ) whereas retaining the option is optimal if S < Sf (τ ). Likewise
for a shout put, the optimal exercise boundary will lie between E ≥ Sf (τ ) ≥ S∗ ,
and early exercise is optimal if S ≤ Sf (τ ).
84
G. ALOBAIDI, R. MALLIER and S. MANSI
At this point, we should say a few words about the analyticity of the free
boundary, and of free boundaries arising in Stefan problems in general. The Stefan problem is concerned with the heat equation with a moving boundary which
is not specified a priori; such problems are typically associated with changes of
phase, such as melting and solidification. Because the Black-Scholes PDE can be
transformed into the heat equation by a change of variables, the free boundary
problems arising in the pricing of options with American-style early exercise features are closely related to the Stefan problem. For many years, the analyticity of
the interfacial boundary arising in the Stefan problem was an issue: for example,
in [36], this was mentioned (in 1971) as a still unsolved problem. The classical
Stefan problem involves heat conduction in a material occupying the semi-infinite
space x > 0 with an arbitrarily prescribed initial temperature UI (x) at τ = 0
for x > 0 and an arbitrarily prescribed boundary condition UB (τ ) at x = 0 for
τ > 0. Because of the change of temperature at x = 0, a new phase of the material
starts to appear, and the phase change occurs along a free boundary x = xf (τ )
where the temperature of both phases is Uf (τ ). If we label the original phase with
the subscript 1 and the new phase with the subscript 2, then the complete set of
equations for the classical Stefan problem is
∂ 2 U1
∂U1
= α1
for x < xf (τ ),
∂τ
∂x2
(8)
2
∂ U2
∂U2
= α2
for x > xf (τ ),
∂τ
∂x2
together with the boundary and initial conditions U1 (0, τ ) = UB (τ ), U2 (x, 0) =
UI (x), and the condition at the free boundary U1 (xf (τ ), τ ) = U2 (xf (τ ), τ ) = Uf (τ )
and a condition on the heat flux at the free boundary,
∂U1 ∂U2 k1
(9)
− k2
= ±ρlx0f (τ ) .
∂x ∂x xf (τ )
xf (τ )
Friedman [18] later showed that if the boundary data UB (τ ) was analytic and
the initial data UI (x) was bounded, and if the initial and boundary data were
continuous at their intersection, then the interfacial boundary was analytic for
τ > 0; however, the analyticity at τ = 0 remained unanswered, because it was
unclear whether xf was a function of τ or τ 1/2 . When xf is a function of τ , it
is analytic at τ = 0, but when it is a function of τ 1/2 , not all derivatives of the
function exist at τ = 0. Tao [40] later extended this analysis by removing the
restriction that the initial and boundary data be continuous at their intersection,
so that he did not require UB (0) = UI (0) as Friedman had. Tao showed that the
interfacial boundary xf (τ ) was an analytic function of τ 1/2 if UB was an analytic
function of τ 1/2 and UI was an analytic function of x. Tao mentioned that his
study was also valid when the effect of density changes during the phase transition
were included, as the convective term which this adds to the governing PDE can
be removed by a change of variable.
In the present problem, we are considering the Black-Scholes-Merton PDE (1),
together with the initial condition that V (S, T ) is specified at t = T , while at
LAPLACE TRANSFORMS AND SHOUT OPTIONS
85
the free boundary we have V (S, t) = Vf (S, t) and (∂V /∂S) = (∂Vf /∂S). We
note that early exercise is possible at any time rather than on discrete occasions like a Bermudan and that the pay-off from early exercise is smooth. If
we make the transformation V (S, t) = eγx+δτ U (x, τ ) + Vf (S, t), with x = ln(S/E),
τ = T − t, γ = 1/2 − (r − D)/σ 2 and δ = −r − γ 2 σ 2 /2, then we find that U obeys
a nonhomogeneous heat equation of the form
(10)
∂U
σ2 ∂ 2 U
+ Uf (x, τ ) ,
=
∂τ
2 ∂x2
together with the condition that U (x, 0) is specified at τ = 0 while at the free
boundary we have U (x, t) = (∂Uf /∂x) = 0. This transformation enables us to
carry over many of the results on the Stefan problem given in [18, 40], and conclude that the location of the free boundary Sf (τ ) is analytic in τ 1/2 for τ > 0 while
not all derivatives of Sf (τ ) exist at τ = 0. This result also holds for vanilla American options, and indeed in that case,
it is known that
i close to expiry the free boundh p
ary behaves like Sf (τ ) ∼ S0 exp x1 σ 2 τ /2 + · · · for the vanilla call with D < r
h
i
(0) √
and the vanilla put with D > r, and like Sf (τ ) ∼ S0 exp x1
−τ ln τ + · · · for
the vanilla call with D > r and the vanilla put with D < r [14, 6, 2, 3, 17], so
that the derivatives Sf0 (τ ), Sf00 (τ ), · · · , do not exist at τ = 0. Similarly, for the
shout options considered here, when we come to use asymptotics to examine the
behavior of the free boundary close to expiry, we will see that the slope of the
boundary is infinite right at expiry; this does not affect our analysis. As discussed
above, we also expect that Sf (τ ) asymptote to S∗ as τ → ∞, so that Sf0 (τ ) → 0 in
that limit. We should mention that, in addition to the studies on the analyticity of
the free boundary for the Stefan problem, several researchers have looked at similar issues for the free boundary for vanilla American options. Van Moerbeke [42],
whose study was contemporaneous with [18] and pre-dated that of [40], showed
that the free boundary Sf (τ ) for American options was continuously differentiable;
it should be possible to apply the results of [42] to the shout options considered
here. Karatzas [20, 15] was able to prove the existence of an optimal exercise
policy for American options and show that there was an optimal stopping time.
A number of researchers, for example [19, 21, 9], have studied American options
by decomposing them into a European option together with an early exercise premium. In a sense, shout options can be viewed as supercharged American options,
since they have the same pay-off at expiry but a higher pay-off for early exercise,
and therefore the same sort of decomposition should work for shout options, and
therefore some of the theoretical results embedded in [19, 21, 9] should carry over
to shout options.
In our analysis, we have also used τf (S), the inverse function of Sf (τ ), which
represents the location of the free boundary as a function of the stock price. Since
Sf (τ ) is analytic away from τ = 0 and is a monotone function, the inverse τf (S)
will be analytic also on the interval E < S < S∗ for the call and S∗ < S < E for
the put. At the ends of this interval, we have τf → ∞ as S → S∗ , and τf → 0 as
86
G. ALOBAIDI, R. MALLIER and S. MANSI
S → E. Once again, the behavior at the ends of this interval does not affect our
analysis.
2.2. Partial Laplace transform in time
Having formulated the problem, we shall now attempt to solve it using a Laplace
transform in time. This is the same technique we used for American options in
[24], and, as in that study, since the Black-Scholes-Merton PDE only holds where
it is optimal to retain the option, we will modify the usual definition
Z∞
(11)
L(G)(p) = g(τ ) e−pτ dτ
0
somewhat, and define our version as follows for S ≤ S∗ for the call and S ≥ S∗ for
the put,
Z∞
(12)
V(S, p) =
V (S, τ ) e−pτ dτ,
τf (S)
so that the lower limit is τ = τf (S) rather than τ = 0. As mentioned in §1, the
partial Laplace transform has been used to successfully tackle diffusion problems
in the past, notably by [16]. Returning to our definition of the Laplace transform,
this is of course equivalent to setting V (S, τ ) = 0 in the region where it is optimal
to exercise. This means the inverse is only meaningful where it is optimal to retain
the option. Because of this definition, the price of the option V (S, τ ) will obey
the equation (2) everywhere where we integrate. We require the real part of p to
be positive for the integral in (12) to converge. In addition, we know from the
definition that V(S, p) → 0 as S → S∗ . We can also define an inverse transform
(13)
1
V (S, τ ) =
2π i
γ+i
Z ∞
V(S, p) epτ dp.
γ−i ∞
Given our definition of the forward transform, this inverse is only meaningful where
it is optimal to hold the option. From our definition, the following transforms can
be derived easily,
∂V
L
= pV − e−pτf (S) Vf (S, τf (S)),
∂τ
∂V
dV
L
=
+ e−pτf (S) τf0 (S)Vf (S, τf (S)),
(14)
∂S
dS
2 ∂ V
d
∂V
∂Vf
L
=
L
+ e−pτf (S) τf0 (S)
(S, τf (S)).
2
∂S
dS
∂S
∂S
In the above, we have adopted the convention that, for the call, τf (S) is the
location of the free boundary for E < S < S∗ , but for S < E, we set τf = 0 since
it is optimal to hold the option to expiry, while for the put τf (S) is the location
of the free boundary for E > S > S∗ , but for S > E, we set τf = 0 since it is
LAPLACE TRANSFORMS AND SHOUT OPTIONS
87
optimal to hold the option to expiry. Applying this partial Laplace transform to
the governing PDE (2), we arrive at the following (nonhomogeneous Euler) ODE
for the transform of the option price,
2 2 2
σ S d
d
+
(r
−
D)
S
(15)
−
(p
+
r)
V + F (S) = 0,
2 dS 2
dS
where the nonhomogeneous term F (S) takes a different value in various regions.
For the shout option, we have two separate regions:
Region (a): 0 < S < E for the call and S > E for the put, where we have
V (Sf (τ ), τ ) = 0, ∂V
∂S (Sf (τ ), τ ) = 0, τf = 0 and
(16)
F (S) = 0.
Region (b): E < S < S∗ for the call and E > S > S∗ for the put, where
∂Vf
V (Sf (τ ), τ ) = Vf (Sf (τ ), τ ), ∂V
∂S (Sf (τ ), τ ) = ∂S (Sf (t), t), τf > 0 and
(17)
F (S) = e−pτf (S) [F0 (S) + (p + p0 ) F1 (S)] ,
2 2
∂
∂
σ S
τf00 (S) + p0 τf02 (S) + τf02 (S)
+ 2τf0 (S)
F0 (S) =
2
∂τ
∂S
0
+1 + (r − D) Sτf (S) Vf (S, τf (S)),
F1 (S) = −
σ 2 S 2 02
τ (S)Vf (S, τf (S)),
2 f
where Vf (S, t) was given in (3) for the call and in (4) for the put, and F0 and F1
are introduced to simplify the Laplace inversion later, and
p0 =
4(D − r)2 + 4σ 2 (D + r) + σ 4
.
8σ 2
The general solution of (15) is
Z
2
2
1
1
2
S − 2σ2 (2D−2r+3σ +λ(p)) F (S)dS
V = S 2σ2 (2D−2r+σ +λ(p)) C1 (p) −
λ(p)
Z
2
2
1
1
2
S − 2σ2 (2D−2r+3σ −λ(p)) F (S)dS ,
(18) + S 2σ2 (2D−2r+σ −λ(p)) C2 (p) +
λ(p)
1/2
where λ(p) = 23/2 σ [p + p0 ] , and C1 and C2 are constants of integration, which
may depend on the transform variable p. Notice that since r, D and σ are all
assumed to be positive, and we assume that p has a positive real part from
the definition of the Laplace
transform, then the real part of the first exponent
2D − 2r + σ 2 + λ(p) /(2σ 2 ) is assumed positive, while the real part of the second
exponent, 2D − 2r + σ 2 − λ(p) /(2σ 2 ) is assumed negative.
2.3. Analysis for the call
Considering first the call, applying this solution (18) to the two separate regions
outlined above, we find that in region (a) we must discard the second solution in
88
G. ALOBAIDI, R. MALLIER and S. MANSI
order to satisfy the boundary condition on S = 0 that V (0, t) = 0 and consequently
that V(0, p) = 0, so in this region we have
V=
(19)
(a)
C1 (p)
S
E
1
2σ 2
(2D−2r+σ2 +λ(p))
.
In region (b), we find the solution which satisfies the condition that V(S, p) → 0
as S → S∗ is
2
V=
Sλ(p)
ZS∗
Se
S
!−
1
2σ 2
(2D−2r+3σ2 )
S
(20)

Se
×
S
!− λ(p)2
2σ
−
Se
S
! λ(p)2 
2σ
e S,
e
 F (S)d
where F (S) is given by (17). We must now match the solutions in these two
regions together. We require that V and (dV/dS) are continuous across S = E,
which tells us that
(21)
(a)
C1 (p)
2
=
Eλ(p)
ZS∗ − 1 2D−2r+3σ2 +λ(p)
(
)
2σ 2
e S,
e
Se
F (S)d
E
and
Z
S∗
(22)
1
Se− 2σ2 (2D−2r+3σ
2
−λ(p))
e Se = 0,
F (S)d
E
where F (S) is given by (17). This equation is of the same form as that for the
American call with D > r given in [24], but of course the nonhomogeneous term
F (S) is different since the pay-off at early exercise is different. For the shout
options discussed here, the pay-off from shouting is given by (3) and (4), whereas
for vanilla American options, the terms involving erfc in (3) and (4) would be
absent.
Of course, (22) is actually the Laplace transform of an integro-differential equation in (S, τ ) space, and we can obtain this latter equation by applying the inverse Laplace transform (13) to (22). To invert the transform, we first divide by
2
2
2
(p + p0 )3/2 E −(2D−2r+3σ )/(2σ ) (Sf (τ ))λ(p)/(2σ ) , and rewrite (22) as
ZS∗
(23)
!−
1
2σ 2
(2D−2r+3σ2 )
" p
#
2(p + p0 ) Sf (τ )
exp −
ln
σ
Se
E
"
#
e
e
e−pτf (S) F0 (S)
e dSe = 0,
×√
+ F1 (S)
p + p0 p + p0
Se
E
LAPLACE TRANSFORMS AND SHOUT OPTIONS
89
and then use the following standard inverse transforms [35],
L−1 e−ap G(p) = H (τ − a) g (τ − a) ,
L−1 [G(p + p0 )] = e−p0 τ g(τ ),
2
h
i
a
1
L−1 p−1/2 exp −ap1/2 = √ 1/2 exp −
,
4τ
πτ
2
i 2τ 1/2
h
a
a
−3/2
1/2
−1
= √ exp −
p
exp −ap
L
− a erfc √ ,
4τ
π
2 τ
(24)
where H(t) is the Heaviside step function, to obtain
!− 12 (2D−2r+3σ2 )
ZS∗ 2σ
q
e
S
e
e
e
H τ − τf (S)
τ − τf (S)
e−p0 (τ −τf (S))
E
E
 
2 
!
e
ln(S
(τ
)/
S)
e
f


 1
e + F1 (S)
(25)
2F0 (S)
exp −
× √

2
e
e
π
τ − τf (S)
2σ (τ − τf (S))


√
e
e
e
2F0 (S) ln(Sf (τ )/S)
ln(Sf (τ )/S)  e
−
erfc  q
dS = 0,
e
σ(τ − τf (S))
e
σ 2(τ − τf (S))
or applying the step function
SZ
f (τ )
q
e
τ − τf (S)
Se
E
!−
1
2σ 2
(2D−2r+3σ2 )
e−p0 (τ −τf (S))
e
E

(26)
 1
× √
π
e
e + F1 (S)
2F0 (S)
e
τ − τf (S)
!
 
exp −
e
ln(Sf (τ )/S)
2 


e
− τf (S))


√
e
e
e
2F0 (S) ln(Sf (τ )/S)
ln(Sf (τ )/S)  e
−
erfc  q
dS = 0.
e
σ(τ − τf (S))
e
σ 2(τ − τf (S))
2σ 2 (τ
To evaluate this, we will change variables so that we have an integral with respect
to τ ,
− 12 (2D−2r+3σ2 )
Z τ
2σ
√
Sf (z)
e−p0 (τ −z)
τ −z
E
0
"
!
!
2
1
Fe1 (z)
(ln(Sf (τ )/Sf (z))
e
× √
2F0 (z) +
exp −
(27)
τ −z
2σ 2 (τ − z)
π
!#
√
2Fe0 (z) ln(Sf (τ )/Sf (z))
ln(Sf (τ )/Sf (z))
p
−
erfc
dz = 0,
σ(τ − z))
σ 2(τ − z)
90
G. ALOBAIDI, R. MALLIER and S. MANSI
where
Fe0 (τ ) = F0 (Sf (τ ))Sf0 (τ )
"
=
Sf0 (τ )
(28)
+
+ r−D+σ
2
σ 2 Sf2 (τ )
Sf (τ ) +
2Sf0 (τ )
Sf00 (τ )
p0 − 0
Sf (τ )
!#
Vf (Sf (τ ), τ )
σ 2 Sf2 (τ ) ∂Vf
(Sf (τ ), τ ) + σ 2 Sf (τ )E,
2Sf0 (τ ) ∂τ
Fe1 (τ ) = F1 (Sf (τ ))Sf0 (τ )
=−
σ 2 Sf2 (τ )
Vf (Sf (τ ), τ ).
2Sf0 (τ )
This last equation (27) is an integro-differential equation in (S, τ ) space for the
location of the free boundary for the shout call. By making the substitutions
Sf (τ ) = E exf (τ ) and z = τ y, we can rewrite this as
Z1 p
0
1
1 − y exp − 2 2D − 2r + 3σ 2 xf (yτ ) − p0 τ (1 − y)
2σ
"
!
!
2
e1 (yτ )
F
1
(x
(τ
)
−
x
(yτ
))
f
f
(29)
2Fe0 (yτ ) +
× √
exp −
τ (1 − y)
2σ 2 τ (1 − y)
π
!#
√
Fe0 (yτ ) 2(xf (τ ) − xf (yτ ))
(xf (τ ) − xf (yτ ))
p
erfc
dy = 0.
−
στ (1 − y))
σ 2τ (1 − y)
2.3.1. The free boundary close to expiry. While we have been unable to
obtain a complete solution to (29), and such a solution would almost certainly
need to numerical, it is possible to study the behavior of the free boundary close
to expiry, in the limit τ → 0, by making several approximations and assumptions.
We will assume that in this limit, the free boundary behaves like [14]
p
xf (τ ) ∼ x1 σ 2 τ /2 + x2 τ + · · · ,
(30)
and substitute this assumed form into the
p integro-differential equation (29), which
we expand as a series in τ . The factor σ 2 /2 is included to simplify the analysis
at a later stage. Upon making this substitution, we will attempt to solve for the
coefficient x1 and either of two things can happen: if the equation for x1 has no
solution or the solution is clearly wrong, for example of the wrong sign, then we
must conclude that our ansatz (30) is incorrect, while on the other hand if we are
able to find a plausible value for x1 , we can conclude that the expansion (30) is
correct with the leading coefficient x1 as given.
The assumed form (30) was chosen because it is both the simplest and the most
common form found in this kind of free boundary problem, and is the form of
the boundary for the vanilla American call with D < r and the vanilla American
put with D > r. We note that in a companion paper [5] on another exotic, the
installment option, we used a similar approach, and found that the ansatz (30)
LAPLACE TRANSFORMS AND SHOUT OPTIONS
91
was unsuccessful for installment options, but that our second guess, an ansatz of
(0) √
the form xf (τ ) ∼ x1
−τ ln τ + · · · , would work: this is also the form of the
free boundary for for the vanilla call with D > r and the vanilla put with D < r
[14, 6, 2, 17].
Returning to the shout call, in the limit τ → 0, we can show that for the
assumed form (30), at leading order Fe0 and Fe1 behave like
√ √
E 2 σ2
√ 1 + 3x1 π + x21 + 2x31 π + O (τ ) ,
Fe0 (τ ) ∼
2x1 π
√ E 2 σ2
√ 1 + 2x1 π τ + O τ 2 ,
Fe1 (τ ) ∼ −
2x1 π
(31)
and upon expanding (29) as a series in τ , at leading order we require that
Z1 p
1−y
0
√
√ (1 + 2x1 π) y
√
2
3
2 1 + 3x1 π + x1 + 2x1 π −
1−y
(32)
!
√
2
x2 1 − y
× exp − 1
1−y
√
√ √
√ #
2 1 + 3x1 π + x21 + 2x31 π x1 (1 − y)
x1 (1 − y)
√
−
erfc
dy = 0,
1−y
1−y
1
× √
π
which is the equation we must solve for x1 . To date, we have been unable to find a
closed form expression for x1 , but evaluating (32) numerically with the computer
algebra package Maple returns a value of x1 ≈ 0.160536690345, so for the shout
call the behavior of the free boundary close to expiry is given by
i
h
p
Sf ∼ E exp 0.160536690345 σ 2 τ /2 + O (τ )
h
i
(33)
∼ E exp 0.1135166στ 1/2 + O (τ )
h
i
∼ E exp 0.1135166σ(T − t)1/2 + O (T − t) .
At leading order, this does not depend on r or D but does depend on σ. It is worth
recalling that the corresponding result for a vanilla American call is x1 = 0.903447
[14, 2] if D < r while is D ≥ r, the assumed form (30) must be replaced by one
involving logarithms. This means that close to expiry, the free boundary for a
shout call is less steep than that for a vanilla American call, which presumably is
due to early exercise being more attractive for a shout than a vanilla American
because of the sweeter pay-off.
2.3.2. Laplace inversion. With regard to the value of the option itself, we can
also apply an inverse Laplace transform (13) to the expressions (19), (20) for
92
G. ALOBAIDI, R. MALLIER and S. MANSI
V(S, p) to obtain expressions for V (S, τ ). To do this, we make use of (24) together
with
2
i
h
a2 − 2τ
a
(34)
L−1 p1/2 exp −ap1/2 = √ 5/2 exp −
.
4τ
4 πτ
For 0 < S < E, we can use (21) to invert (19), proceeding in the same manner as
we did in (23–26) to obtain
(35)
SZ
f (τ )
!− 12 (2D−2r+3σ2 )
e
2σ
e−p0 (τ −τf (S)) Se
q
S
e
τ − τf (S)
E
2 
e
ln(
S/S)


× exp −

2
e
2σ (τ − τf (S))
 

2
e
ln(
S/S)
1
 e

e + F1 (S)
e 
× F0 (S)
−
 dS
 2
2
e
e
2σ (τ − τf (S))
2(τ − τf (S))
1
V = √
2πσS

Zτ
− 12 (2D−2r+3σ2 )
2σ
Sf (z)
S
0
!
2
(ln(Sf (z)/S))
× exp −
2σ 2 (τ − z)
"
!#
2
(ln(S
(z)/S))
1
f
× Fe0 (z) + Fe1 (z)
−
ddz,
2σ 2 (τ − z)2
2(τ − z)
= √
1
2πσS
e−p0 (τ −z)
√
τ −z
which is an expression for the value of the option for 0 < S < E. For E < S < S∗ ,
the inversion is a little more complicated. We can invert (20) directly to obtain
!− 12 (2D−2r+3σ2 )
√ ZS∗
2σ
2
Se
e
e e−p0 (τ −τf (S))
V =−
H(τ − τf (S))
σS
S
S
"√ √
#!
e
e + pF1 (S)
e
2
p
ln(
S/S)
F
(
S)
0
e τ − τf (S))d
e Se
× L−1
sinh
(S,
√
p
σ
(36)
!− 12 (2D−2r+3σ2 )
SZf (t)
√
2σ
2H(τ − τf (S))
Se
e
e e−p0 (τ −τf (S))
=−
H(τ − τf (S))
σS
S
S
"√ √
#!
e
e + pF1 (S)
e
2 p ln(S/S)
F0 (S)
−1
e τ − τf (S))d
e S,
e
×L
sinh
(S,
√
p
σ
which is useful in the sense that it shows that V (S, τ ) must vanish if τ < τf (S),
meaning in the region where exercise is optimal, which was expected given our
LAPLACE TRANSFORMS AND SHOUT OPTIONS
93
definition of the partial Laplace transform (12). To proceed further, it is necessary
to rewrite the sinh function in (36) in terms of exponentials and then use (22).
For τ > τf (S) and E < S < S∗ , we then have
S
Z ∗ e !− 2σ12 (2D−2r+3σ2 ) −pτf (S)
e
1
S
e
−1 
√
√
V =
L
S
p + p0
2σS
S
!
"√ √
#
i
e h
2 p + p0 ln(S/S)
e
e
e
F0 (S) + (p + p0 ) F1 (S) dS
× exp
σ
 S
!− 12 (2D−2r+3σ2 )
Z
e
2σ
1
e−pτf (S)
Se
−1 
√
+√
L
S
p + p0
2σS
E
"√ √
#
!
h
i
e
2 p + p0 ln(S/S)
e
e
e
× exp
F0 (S) + (p + p0 ) F1 (S) dS
σ
!− 12 (2D−2r+3σ2 )
ZS∗ H τ − τf (S)
e
2σ
1
Se
e
q
e−p0 (τ −τf (S))
= √
S
2πσS
e
τ − τf (S)
E

2 
e
ln(S/S)


× exp −

2
e
2σ (τ − τf (S))
(37)

 
2
e
ln(S/S)
1

 e
e + F1 (S)
e 
× F0 (S)
−
 2
 dS
e 2
e
2σ (τ − τf (S))
2(τ − τf (S))
= √
1
2πσS
SZ
f (τ )
e
−p0 (τ −τf (S))
e
q
Se
S
!−
1
2σ 2

2 
(2D−2r+3σ2 )
e
ln(
S/S)


× exp −

2
e
2σ (τ − τf (S))

e
τ − τf (S)

 2
e
ln(
S/S)
1

 e
e + F1 (S)
e 
× F0 (S)
−
 dS
 2
e 2
e
2σ (τ − τf (S))
2(τ − τf (S))
E
!
− 12 (2D−2r+3σ2 )
Zτ −p0 (τ −z) 2
2σ
1
e
Sf (z)
(ln(Sf (z)/S))
√
= √
× exp −
S
2σ 2 (τ − z)
τ −z
2πσS
0
"
!#
2
(ln(Sf (z)/S))
1
e
e
× F0 (z) + F1 (z)
−
dz,
2σ 2 (τ − z)2
2(τ − z)
while for τ < τf (S) and E < S < S∗ we have V (S, τ ) = 0. It is worth noting
that the expression (35) for 0 < S < E and (37) for E < S < S∗ are the same.
This expression (37) for the option value V (S, τ ) involves the location of the free
94
G. ALOBAIDI, R. MALLIER and S. MANSI
boundary Sf (τ ) and its derivatives. Other than very close to expiry, the evaluation
of (37), would have to be implemented numerically, after first having computed
Sf (z) on the interval 0 < z < τ using (27).
2.4. Analysis for the put
Turning now to the put, once again, we apply the solution (18) to the two separate
regions defined earlier, and now we find that in region (a) we must discard the
first solution in order to satisfy the boundary condition that V → 0 as S → ∞, so
in this region we have
12 (2D−2r+σ2 −λ(p))
S 2σ
(a)
(38)
.
V = C2 (p)
E
In region (b), we find the solution which satisfies the condition that V(S, p) → 0
as S → S∗ is
Z S∗ e !− 2σ12 (2D−2r+3σ2 )
2
S
V=
Sλ(p) S
S

(39)
! λ(p)2
!− λ(p)2 
2σ
2σ
e
e
S
S
e S,
e
 F (S)d
−
×
S
S
where F (S) is again given by (17), but of course Vf is different, being given by
(3) rather than (4). If we compare these expressions to those for the call (19,20),
they appear more alike than different. One of the differences is that in region (a),
for the call, we discarded the second homogeneous solution given in (18), while
for the put, we discarded the first. Another is that in region (b), the solutions for
the call and put have opposite signs, because the region is E < S < S∗ for the
call and S∗ < S < E for the put. We must now match the solutions in these two
regions together, and as we did for the call, we will require that V and (dV/dS)
be continuous across S = E. Proceeding as we did for the call, we find that
ZE e !− 2σ12 (2D−2r+3σ2 −λ(p))
2
S
(a)
e S,
e
C2 (p) =
(40)
F (S)d
Eλ(p)
E
S∗
which is similar to its counterpart (21) for the call, except that the subscripts 1
and 2 are interchanged and the sign of λ(p) in the exponent is reversed, and
ZE
(41)
1
Se− 2σ2 (2D−2r+3σ
2
+λ(p))
e Se = 0,
F (S)d
S∗
where again F (S) is given by (17). This integro-differential equation for the put
bears a strong resemblance to the equation for the call (22), with at first glance
only the sign of λ(p) being different, although of course the nonhomogeneous term
F (S) which appears in these is slightly different for the put than for the call: the
definition (17) is the same for both cases, but of course Vf is given by (3) for
95
LAPLACE TRANSFORMS AND SHOUT OPTIONS
the call and by (4) for the put. Once again, this equation is also very similar to
the corresponding equation for the American put given in [24], with of course the
nonhomogeneous term F (S) again being different.
As was the case with (22), (41) is the Laplace transform of an integro-differential
equation in (S, τ ) space, and once again we can apply the inverse Laplace transform
(13) to obtain a second integro-differential equation. To invert the transform, we
2
2
2
again divide by (p + p0 )3/2 E −(2D−2r+3σ )/(2σ ) (Sf (τ ))−λ(p)/(2σ ) , and rewrite (41)
as
ZE
(42)
Se
E
!−
1
2σ 2
(2D−2r+3σ2 )
S∗
"p
#
2(p + p0 )
Se
exp
ln
σ
Sf (τ )
"
#
e
e
e−pτf (S) F0 (S)
e dSe = 0 ,
×√
+ F1 (S)
p + p0 p + p0
and then use the standard inverse transforms (24)to obtain
ZE
q
e
e
τ − τf (S)
H τ − τf (S)
Se
E
!−
1
2σ 2
(2D−2r+3σ2 )
e−p0 (τ −τf (S))
e
S∗

(43)
 1
× √
π
e
e + F1 (S)
2F0 (S)
e
τ − τf (S)
 !

exp −
e f (τ ))
ln(S/S
2 


e
− τf (S))


√
e 2 ln(S/S
e f (τ ))
e f (τ ))
F0 (S)
ln(
S/S
 dSe = 0,
−
erfc  q
e
σ(τ − τf (S))
e
σ 2(τ − τf (S))
2σ 2 (τ
or applying the step function
ZE q
e
τ − τf (S)
Se
E
!−
1
2σ 2
(2D−2r+3σ2 )
e−p0 (τ −τf (S))
e
Sf (τ )

(44)
 1
× √
π
e
e + F1 (S)
2F0 (S)
e
τ − τf (S)
!
 
exp −
e f (τ ))
ln(S/S
2σ 2 (τ
2 
e
− τf (S))




√
e
e
e
F0 (S) 2 ln(S/Sf (τ ))
ln(S/Sf (τ ))  e
−
erfc  q
dS = 0.
e
σ(τ − τf (S))
e
σ 2(τ − τf (S))
96
G. ALOBAIDI, R. MALLIER and S. MANSI
To evaluate this, we will change variables so that we have an integral with respect
to τ ,
Zτ
√
0
(45)
− 12 (2D−2r+3σ2 )
2σ
Sf (z)
τ −z
e−p0 (τ −z)
E
!
!
"
2
e1 (z)
F
(ln(S
(z)/S
(τ
))
1
f
f
2Fe0 (z) +
exp −
× √
τ −z
2σ 2 (τ − z)
π
!#
√
Fe0 (z) 2 ln(Sf (z)/Sf (τ ))
ln(Sf (τ )/Sf (z))
p
−
erfc
dz = 0,
σ(τ − z))
σ 2(τ − z)
where
Fe0 (τ ) = F0 (Sf (τ ))Sf0 (τ )
"
=
(46)
Sf0 (τ )
+
+ r−D+σ
2
σ 2 Sf2 (τ )
Sf (τ ) +
2Sf0 (τ )
Sf00 (τ )
p0 − 0
Sf (τ )
!#
Vf (Sf (τ ), τ )
σ 2 Sf2 (τ ) ∂Vf
(Sf (τ ), τ ) − σ 2 Sf (τ )E,
2Sf0 (τ ) ∂τ
Fe1 (τ ) = F1 (Sf (τ ))Sf0 (τ )
= −
σ 2 Sf2 (τ )
Vf (Sf (τ ), τ ).
2Sf0 (τ )
As with (27), this last equation (45) is an integro-differential equation in (S, τ )
space for the location of the free boundary, this time for the shout put. It differs
from (27) in that the ratio Sf (z)/Sf (τ ) is reversed, and also Fe0 and Fe1 will be
slightly different because the pay-off at shouting for the put differs from that for
the call. By making the substitutions Sf (τ ) = E exf (τ ) and z = τ y, we can rewrite
(45) as
Z1 p
0
(47)
1
1 − y exp − 2 2D − 2r + 3σ 2 xf (yτ ) − p0 τ (1 − y)
2σ
"
!
!
2
e1 (yτ )
1
F
(x
(yτ
)
−
x
(τ
))
f
f
× √
2Fe0 (yτ ) +
exp −
τ (1 − y)
2σ 2 τ (1 − y)
π
!#
√
Fe0 (yτ ) 2(xf (yτ ) − xf (τ ))
(xf (yτ ) − xf (τ ))
p
−
erfc
dy = 0.
στ (1 − y))
σ 2τ (1 − y)
2.4.1. The free boundary close to expiry. Once again, we will assume that in
the limit τ → 0, the free boundary has the form (30), and substitute this assumed
form into the integro-differential equation (47), which we expand as a series in τ .
LAPLACE TRANSFORMS AND SHOUT OPTIONS
97
In this limit, we can show that at leading order,
√ √
E 2 σ2
√ 1 + x1 π + x21 − 2x31 π + O (τ )
Fe0 (τ ) ∼
2x1 π
√ E 2 σ2
√ 1 − 2x1 π τ + O τ 2 .
Fe1 (τ ) ∼ −
2x1 π
(48)
The coefficients in (48) for the put differ slightly from their counterparts for the
call (31). Upon expanding (47) as a series in τ , at leading order we require that
Z1 p
1−y
0
√
√
√ (1 − 2x1 π) y
1
× √
2 1 + x1 π + x21 − 2x31 π −
1−y
π
(49)
√ 2 !
2
x 1− y
× exp − 1
1−y
√
√ √
√ #
2 1 + x1 π + x21 − 2x31 π x1 (1 − y)
x1 (1 − y)
√
−
erfc
dy = 0,
1−y
1−y
which is the equation we must solve for x1 for the put. As with (32) for the call,
we evaluated (49) numerically and found a value of x1 ≈ −0.745457861349, so for
the shout put the behavior of the free boundary close to expiry is given by
h
i
p
Sf ∼ E exp −0.745457861349 σ 2 τ /2 + O (τ )
h
i
(50)
∼ E exp −0.5271183στ 1/2 + O (τ )
h
i
∼ E exp −0.5271183σ(T − t)1/2 + O (T − t) .
As expected, the coefficient x1 was positive for the call but negative for the put.
Once again, we can compare the value for x1 with that for a vanilla American. For
a vanilla American put, if D > r we can deduce that x1 = −0.90345 using put-call
symmetry [25, 10], while if D ≤ r once again the series (30) must be replaced by
one including logs [6, 23, 3]. In either case, the boundary for the shout put is not
as steep close to expiry as that for the American put, again because the shout put
is more likely to be exercised early.
2.4.2. Laplace inversion. As for the call, we can apply an inverse Laplace transform to the expressions (38), (39) for V(S, p) to obtain V (S, τ ) for the put, with
the analysis extremely similar to that for the call presented earlier. For τ < τf (S),
once again we find that V (S, τ ) = 0, which again was to be expected given the
definition of our partial Laplace transform (12), while for τ > τf (S), we find for
98
G. ALOBAIDI, R. MALLIER and S. MANSI
both S∗ < S < E and S > E that
(51)
SZ
f (τ )
e − 12 (2D−2r+3σ 2 )
2σ
e−p0 (τ −τf (S)) S
q
Se
e
τ − τf (S)
E
2 
e
ln(
S/S)


× exp −

2
e
2σ (τ − τf (S))

 
2
e
ln(S/S)
1

 e
e + F1 (S)
e 
× F0 (S)
−
 2
 dS
e 2
e
2σ (τ − τf (S))
2(τ − τf (S))
1
V = √
2πσS

1
= −√
2πσS
Zτ
e−p0 (τ −z)
√
τ −z
0
× exp −
(ln(Sf (z)/S))
2σ 2 (τ − z)
"
× Fe0 (z) + Fe1 (z)
2
S
Sf (z)
!
−
1
2σ 2
2
(2D−2r+3σ2 )
(ln(Sf (z)/S))
1
−
2σ 2 (τ − z)2
2(τ − z)
!#
dz ,
which is extremely similar to its counterpart (37) for the call. Other than very
close to expiry, the evaluation of (51), just like that of (37), would have to be
implemented numerically, after first having computed Sf on the interval 0 < z < τ
using (45).
3. Discussion
In this paper we have used Laplace transform methods to study shout options.
The put and call shouts lead to very similar problems in spite of their well-known
important differences. The integro-differential equations (22), (41) and (29), 47
presented above form the main result of this paper. Each of the first set (22), (41)
is a nonlinear (Fredholm) integro-differential equation for the location of the free
boundary, τf (S), or more specifically, an Urysohn equation of the first kind [35].
In this context, Fredholm simply means that the upper and lower limits of the
integral are both constants [33, 34]. The second set of equations (29), (47) were
derived by applying an inverse transform to (22), (41) and involve Sf (τ ) rather
than τf (S),
The equation for the call (22) differs slightly from that for the put (41), since the
pay-offs differ. However, the differences are very small, the main difference being
that λ(p) is replaced by −λ(p). The behavior seems symmetric between some of
the intermediate equations, such as (19), (38), while in others such as (20), (39) it
is anti-symmetric. Presumably there must be some sort of put-call symmetry for
LAPLACE TRANSFORMS AND SHOUT OPTIONS
99
shout options, along the same lines as that for vanilla American options [10, 25],
although it is not immediately obvious what form that symmetry takes.
We note that each of our integro-differential equations (22), (41) involves the
transform variable p, both through the exponent of S and also through the term
e−pτf (S) in F (S). Since τf (S) is a physical quantity, obviously it must be independent of p, and so the solution of each of these integro-differential equations involves
finding a function τf (S) which satisfies the equation for all values of p with a positive real part. Because of this, we can think of each of these integro-differential
equations as being a form of integral transform operating on τf (S), and inverting this integral transform would give us the location of the free boundary τf (S).
However, this inversion would appear to be extremely difficult to do analytically
because of the factor e−pτf (S) in F (S) as given in (17); if this term were absent, we
could regard the equation as a form of (finite) Mellin transform [38, 29], although
not of Naylor-type [30, 31]. While we may not have been able to invert this Mellinlike transform, we were able to apply a standard inverse Laplace transform (13)
to the integro-differential equations (22), (41) to obtain a second set of integrodifferential equations (29), (47) and use this second set to
find the behavior of
the free boundary close to expiry: for the call, Sf ∼ E exp 0.1135166σ(T − t)1/2
while for the put, Sf ∼ E exp −0.5271183σ(T − t)1/2 . It is interesting to compare this behavior to that of the vanilla Americans. For the shout options, the free
boundary always starts from the strike price E at expiry, while for the Americans
this only happens for the call with D ≥ r and the put with D ≤ r. In addition,
close to expiry the free boundary for the shouts always behaves like τ 1/2 , while for
the Americans this form of behavior prevails for the call with D < r and the put
with D > r but for the call with D ≥ r and the put with D ≤ r the behavior close
to expiry involves logs [6, 23, 3]. In either case, the free boundary close to expiry
for shout options seems to be less steep than that for vanilla Americans, and it
would seem likely that this is because early exercise is more likely for a shout than
a vanilla American on the same underlying with the same strike, simply because
the rewards for early exercise are greater for a shout than an American. For an
American, early exercise involves a trade-off between receiving the pay-off earlier
and receiving benefits from any further upside, while with a shout early exercise
results in receiving a portion of the pay-off earlier while still benefiting from further upsides. Because of this, it would appear paradoxically that although shout
options are more complex contracts than vanilla Americans, the analysis of shouts
is actually a little simpler than that of Americans, primarily because logs are not
present for the shouts.
Moving on to the issue of the value of the option, in (19), (20), we have a
series of expressions for V(p, S), the transform of the option price V (S, τ ), with
corresponding expressions in (38), (39) for the put. The constants which appear in
these expressions were also given in the previous sections. We were able to apply
an inverse transform (13) to these expressions, to find the value V (S, τ ) of the
option in the region where it is optimal to hold. We note that our expression for
V (S, τ ), which was given in (37) for the call and (51) for the put, involves τf (S),
100
G. ALOBAIDI, R. MALLIER and S. MANSI
the location of the free boundary, which we know only abstractly as the solution
of the applicable integro-differential equation.
In closing, we would like to make suggestions for further work on shout options.
While [8, 43] have made great strides in valuing shouts numerically, relatively little
theoretical work has been done on these options, or indeed on other exotics with
American-style early exercise features, and we would suggest that studies along
the lines of for instance [17, 22, 23, 39] might be worthwhile for these exotics. At
this point, we would like to say a few words about the motivation for reformulating
a free boundary problem like the present one as an integral or integro-differential
equation, regardless of whether such reformulation is achieved by using Laplace
transforms as in the present study or by one of the other methods mentioned
above. The principal advantage of this approach is that, once an equation such as
(27), (45), or those presented in [22, 23, 39], for the location of the free boundary
is derived, the problem of finding location of the free boundary is decoupled from
the problem of valuing the option.
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102
G. ALOBAIDI, R. MALLIER and S. MANSI
G. Alobaidi, Department of Mathematics & Statistics, American University of Sharjah, Sharjah,
United Arab Emirates, e-mail: galobaidi@aus.edu
R. Mallier, Department of Applied Mathematics, University of Western Ontario, London ON
N6A 5B7 Canada, e-mail: rolandmallier@hotmail.com
S. Mansi, Department of Finance, Pamplin School of Business, Virginia Tech, Blacksburg, VA
24060 USA, e-mail: smansi@vt.edu